math::calculus(n) Math math::calculus(n)
NAME
math::calculus - Integration and ordinary differential equations
SYNOPSIS
package require Tcl 8
package require math::calculus 0.5
::math::calculus::integral begin end nosteps func
::math::calculus::integralExpr begin end nosteps expression
::math::calculus::integral2D xinterval yinterval func
::math::calculus::integral3D xinterval yinterval zinterval func
::math::calculus::eulerStep t tstep xvec func
::math::calculus::heunStep t tstep xvec func
::math::calculus::rungeKuttaStep tstep xvec func
::math::calculus::boundaryValueSecondOrder coeff_func force_func leftbnd rightbnd nostep
::math::calculus::solveTriDiagonal acoeff bcoeff ccoeff dvalue
::math::calculus::newtonRaphson func deriv initval
::math::calculus::newtonRaphsonParameters maxiter tolerance
DESCRIPTION
This package implements several simple mathematical algorithms:
o The integration of a function over an interval
o The numerical integration of a system of ordinary differential equations.
o Estimating the root(s) of an equation of one variable.
The package is fully implemented in Tcl. No particular attention has been paid to the
accuracy of the calculations. Instead, well-known algorithms have been used in a straight-
forward manner.
This document describes the procedures and explains their usage.
PROCEDURES
This package defines the following public procedures:
::math::calculus::integral begin end nosteps func
Determine the integral of the given function using the Simpson rule. The interval
for the integration is [begin, end]. The remaining arguments are:
nosteps
Number of steps in which the interval is divided.
func Function to be integrated. It should take one single argument.
::math::calculus::integralExpr begin end nosteps expression
Similar to the previous proc, this one determines the integral of the given expres-
sion using the Simpson rule. The interval for the integration is [begin, end].
The remaining arguments are:
nosteps
Number of steps in which the interval is divided.
expression
Expression to be integrated. It should use the variable "x" as the only
variable (the "integrate")
::math::calculus::integral2D xinterval yinterval func
The command integral2D calculates the integral of a function of two variables over
the rectangle given by the first two arguments, each a list of three items, the
start and stop interval for the variable and the number of steps.
The currently implemented integration is simple: the function is evaluated at the
centre of each rectangle and the content of this block is added to the integral. In
future this will be replaced by a bilinear interpolation.
The function must take two arguments and return the function value.
::math::calculus::integral3D xinterval yinterval zinterval func
The command Integral3D is the three-dimensional equivalent of integral2D. The func-
tion taking three arguments is integrated over the block in 3D space given by three
intervals.
::math::calculus::eulerStep t tstep xvec func
Set a single step in the numerical integration of a system of differential equa-
tions. The method used is Euler's.
t Value of the independent variable (typically time) at the beginning of the
step.
tstep Step size for the independent variable.
xvec List (vector) of dependent values
func Function of t and the dependent values, returning a list of the derivatives
of the dependent values. (The lengths of xvec and the return value of "func"
must match).
::math::calculus::heunStep t tstep xvec func
Set a single step in the numerical integration of a system of differential equa-
tions. The method used is Heun's.
t Value of the independent variable (typically time) at the beginning of the
step.
tstep Step size for the independent variable.
xvec List (vector) of dependent values
func Function of t and the dependent values, returning a list of the derivatives
of the dependent values. (The lengths of xvec and the return value of "func"
must match).
::math::calculus::rungeKuttaStep tstep xvec func
Set a single step in the numerical integration of a system of differential equa-
tions. The method used is Runge-Kutta 4th order.
t Value of the independent variable (typically time) at the beginning of the
step.
tstep Step size for the independent variable.
xvec List (vector) of dependent values
func Function of t and the dependent values, returning a list of the derivatives
of the dependent values. (The lengths of xvec and the return value of "func"
must match).
::math::calculus::boundaryValueSecondOrder coeff_func force_func leftbnd rightbnd nostep
Solve a second order linear differential equation with boundary values at two
sides. The equation has to be of the form (the "conservative" form):
d dy d
-- A(x)-- + -- B(x)y + C(x)y = D(x)
dx dx dx
Ordinarily, such an equation would be written as:
d2y dy
a(x)--- + b(x)-- + c(x) y = D(x)
dx2 dx
The first form is easier to discretise (by integrating over a finite volume) than
the second form. The relation between the two forms is fairly straightforward:
A(x) = a(x)
B(x) = b(x) - a'(x)
C(x) = c(x) - B'(x) = c(x) - b'(x) + a''(x)
Because of the differentiation, however, it is much easier to ask the user to pro-
vide the functions A, B and C directly.
coeff_func
Procedure returning the three coefficients (A, B, C) of the equation, taking
as its one argument the x-coordinate.
force_func
Procedure returning the right-hand side (D) as a function of the x-coordi-
nate.
leftbnd
A list of two values: the x-coordinate of the left boundary and the value at
that boundary.
rightbnd
A list of two values: the x-coordinate of the right boundary and the value
at that boundary.
nostep Number of steps by which to discretise the interval. The procedure returns
a list of x-coordinates and the approximated values of the solution.
::math::calculus::solveTriDiagonal acoeff bcoeff ccoeff dvalue
Solve a system of linear equations Ax = b with A a tridiagonal matrix. Returns the
solution as a list.
acoeff List of values on the lower diagonal
bcoeff List of values on the main diagonal
ccoeff List of values on the upper diagonal
dvalue List of values on the righthand-side
::math::calculus::newtonRaphson func deriv initval
Determine the root of an equation given by
func(x) = 0
using the method of Newton-Raphson. The procedure takes the following arguments:
func Procedure that returns the value the function at x
deriv Procedure that returns the derivative of the function at x
initval
Initial value for x
::math::calculus::newtonRaphsonParameters maxiter tolerance
Set the numerical parameters for the Newton-Raphson method:
maxiter
Maximum number of iteration steps (defaults to 20)
tolerance
Relative precision (defaults to 0.001)
Notes:
Several of the above procedures take the names of procedures as arguments. To avoid prob-
lems with the visibility of these procedures, the fully-qualified name of these procedures
is determined inside the calculus routines. For the user this has only one consequence:
the named procedure must be visible in the calling procedure. For instance:
namespace eval ::mySpace {
namespace export calcfunc
proc calcfunc { x } { return $x }
}
#
# Use a fully-qualified name
#
namespace eval ::myCalc {
proc detIntegral { begin end } {
return [integral $begin $end 100 ::mySpace::calcfunc]
}
}
#
# Import the name
#
namespace eval ::myCalc {
namespace import ::mySpace::calcfunc
proc detIntegral { begin end } {
return [integral $begin $end 100 calcfunc]
}
}
Enhancements for the second-order boundary value problem:
o Other types of boundary conditions (zero gradient, zero flux)
o Other schematisation of the first-order term (now central differences are used, but
upstream differences might be useful too).
EXAMPLES
Let us take a few simple examples:
Integrate x over the interval [0,100] (20 steps):
proc linear_func { x } { return $x }
puts "Integral: [::math::calculus::integral 0 100 20 linear_func]"
For simple functions, the alternative could be:
puts "Integral: [::math::calculus::integralExpr 0 100 20 {$x}]"
Do not forget the braces!
The differential equation for a dampened oscillator:
x'' + rx' + wx = 0
can be split into a system of first-order equations:
x' = y
y' = -ry - wx
Then this system can be solved with code like this:
proc dampened_oscillator { t xvec } {
set x [lindex $xvec 0]
set x1 [lindex $xvec 1]
return [list $x1 [expr {-$x1-$x}]]
}
set xvec { 1.0 0.0 }
set t 0.0
set tstep 0.1
for { set i 0 } { $i < 20 } { incr i } {
set result [::math::calculus::eulerStep $t $tstep $xvec dampened_oscillator]
puts "Result ($t): $result"
set t [expr {$t+$tstep}]
set xvec $result
}
Suppose we have the boundary value problem:
Dy'' + ky = 0
x = 0: y = 1
x = L: y = 0
This boundary value problem could originate from the diffusion of a decaying substance.
It can be solved with the following fragment:
proc coeffs { x } { return [list $::Diff 0.0 $::decay] }
proc force { x } { return 0.0 }
set Diff 1.0e-2
set decay 0.0001
set length 100.0
set y [::math::calculus::boundaryValueSecondOrder coeffs force {0.0 1.0} [list $length 0.0] 100]
KEYWORDS
math, calculus, integration, differential equations, roots
math 1.0 math::calculus(n)